= Acoustic WKB amplitude in an expanding universe
{title2=$\delta_k\propto[a\sqrt{\omega}]^{-1},\quad\omega=kc_s/a$}
For pressure-dominated <linear cosmological density perturbations>, $\ddot\delta_k+2H\dot\delta_k+\omega^2\delta_k=0$. Writing $\delta_k=\chi/a$ gives $\ddot\chi+(\omega^2-\ddot a/a)\chi=0$. In the <WKB approximation>, the oscillation amplitude is $1/(a\sqrt\omega)$. For constant sound speed it is proportional to $a^{-1/2}$; the local frozen-frequency amplitude damping time $H^{-1}$ does not imply a universal global envelope $a^{-1}$.
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